Cross-Channel Color Image Encryption Scheme Based on Discrete Memristive Coupled Neurons and DWT Compression

Abstract

To address the consumption and security of color images for transmission and storage, a cross-channel color image encryption scheme based on a discrete memristive coupled neuron model and DWT compression is designed in this article. Firstly, the dynamics of the discrete memristive coupled neuron system are analyzed and found to possess the hyperchaotic phenomenon, which provides sufficient security for the encryption scheme. Secondly, the color image processed by discrete wavelet transform (DWT) has a quarter of the previous capacity. Then, the color image is combined with a Hash function, and the resulting Hash sequence is given the initial value of the hyperchaotic system. Next, a particle swarm foraging algorithm (PSFA) is designed to better disrupt the correlation in the RGB channel. Finally, a complementary DNA coding rule is implemented for the further encryption of color images. Simulation results show that even with DWT lossy compression, the recovered image can be clearly seen. The performance analysis illustrates that under the hyperchaotic system, the proposed encryption algorithm brings higher security for color images.

1. Introduction

With the advent of the information age, digital images have played an increasingly important role in production and life. However, with the popularization of network communication, more and more image data are facing the serious challenge of security [1,2,3,4,5]. In this context, image encryption technology has become one of the critical means to protect image security. Encryption technology disrupts the structure and changes the content of an image [6,7,8,9,10]. Unauthorized users cannot recover the content and structure of the original image, thus protecting the image information [11,12,13,14,15,16].

In recent years, chaotic systems have been applied in the field of image encryption due to their properties such as high nonlinearity, sensitivity concerning initial conditions and pseudo-randomness [17,18,19,20,21,22]. These properties endow chaotic systems with good randomness and complexity, which can allow them to effectively resist various statistical attacks [23,24,25,26,27,28]. Meanwhile, various types of chaotic systems are fabricated, such as continuous systems [29,30,31,32,33,34,35,36,37,38], discrete systems [39,40,41,42,43] and fractional-order systems [44,45,46,47]. Gao et al. designed a hyperchaotic map and applied it in multi-image encryption, which saved a lot of time for the encryption algorithm [13]. Lai et al. presented a new 1D sine–cosine chaotic map for image encryption and applied randomized DNA encoding rules to encrypt images [48]. M. Naim et al. proposed a new chaotic satellite image encryption algorithm, which uses different filters for filtering and then the Fisher–Yates shuffle rule to disrupt the pixel order. Encryption using the block region approach was achieved to reduce the time [49].

Meanwhile, more and more researchers are considering the issue of image capacity. This is determined through the compression and encryption of the size of the image before it is uploaded to the network. At the recovery end, decryption and image restoration are performed. Wen et al. used a DCT technique to encode and compress the image to extract the DC and AC components. The DC component was discretized using chaotic sequences and the AC component was used for frequency domain reconstruction [50]. Sha et al. proposed a compressible low-cost scheme based on chaotic systems. Firstly, compressed sensing was introduced to address the high data storage factor. Secondly, for grayscale, color and 3D STL graphics were fused into a 3D cube. Finally, an encryption algorithm was used for 3D spatial encryption. This scheme fused multiple image data for processing and greatly improved the transmission variety [51]. Zhang et al. used compression sensing to reduce the image size as well, which reduced obstacles to information transmission and storage [2]. The above scheme was designed to compress the image size for easy image storage and transmission. However, in terms of image reconstruction, compression sensing takes longer. In addition, a color image is worth protecting. The correlation between the pixels of RGB channels in color images is very strong. However, some scholars have not noticed this problem and encrypted the three RGB channels with disruption and diffusion [52]. Based on the above, and to address the color image storage capacity with the strong correlation between pixels of RGB channels, a color image encryption scheme based on discrete memristive coupled neurons and DWT compression is designed in this article. Firstly, the color image is decomposed into four parts via DWT. The LL component includes the contour content of the original image. The LL component is subjected to PSFA scrambling in the sense of disrupting the pixel values across channels. Next, options for adding DNA encoding include addition, XOR, and complementary XOR for the further encryption of the image.

This article achieves the following:

• Phase diagrams, bifurcation diagram, Lyapunov exponents (LEs) and the spectral entropy of the discrete memristive coupled neurons are analyzed;
• The color image is compressed using DWT technology, and it is clearly seen on the reduction side;
• Particle swarm foraging algorithms are designed to disrupt color image planar pixel positions cross-channel;
• Tests and performance analysis are conducted, and the scheme has good security and statistical performance.

2. Basic Theory

2.1. Five-Dimensional Discrete Map

As chaotic systems possess the need for cryptographic foundations and hyperchaotic systems possess more complex dynamical behaviors, this article uses our lab’s chaotic model of discrete memristive coupled neurons [39]. The model is a memristor coupled by Chialvo and Rulkov neurons (C-M-R) and has multiple types of hyperchaotic behavior. The C-M-R mathematical model is shown in Equation (1):

$$\{\begin{array}{l}{x}_1(i+1)={(x_1(i))}^2{e}^{(y_1(i)-x_1(i))}+\delta -k_r(\alpha \cos (\pi z{(i)}^2)+\beta )(x_1(i)-x_2(i))\\ y_1(i+1)=a{y}_1(i)-b{x}_1(i)+c\\ z(i+1)=z(i)+x_1(i)-x_2(i)\\ x_2(i+1)=\frac{\gamma }{1+{(x_2(i))}^2}+y_2(i)+k_r(\alpha \cos (\pi z{(i)}^2)+\beta )(x_1(i)-x_2(i))\\ y_2(i+1)=y_2(i)-u(x_2(i)-\sigma )\end{array}$$

(1)

where x₁(i) and y₁(i) are condition variables of the Chialvo neuron model; x₂(i) and y₂(i) are the condition variables of the Rulkov neuron model. z(i) is the memristor model, which serves as a bridge between the two neurons.

2.1.1. Phase Diagram

The trajectory of a discrete map is iterated through its state variables. The iterative states of the state variables are depicted with the use of phase diagrams.

We set the phase diagram parameters for C-M-R as (ε, k_r, α, β, a, b, γ, μ, σ) = (0.1, 0.3, −0.35, −0.65, 0.7, 0.18, 4.8, 0.01, 1.35), with an initial value of (x₁(1), x₂(1), z(1), x₂(1), y₂(1)) = (−1, −1, 0, 1, 1), the result is shown in Figure 1.

2.1.2. LEs and Bifurcation Diagram

For a discrete system, if one LE’s straight line is greater than 0, it is considered a chaotic state. If two LEs’ lines are greater than 0, the system is considered to be in a hyperchaotic state. The bifurcation diagram clearly shows whether the system is in a periodic or chaotic state. LEs and bifurcation diagrams are complementary to each other. The test is performed on parameter a and parameter b. The result is shown in Figure 2, which clearly shows that the two LEs’ lines are greater than 0 and the bifurcation diagram is in a chaotic state. It can be concluded from the above that the model is hyperchaotic with this parameter and its dynamics are more complex and suitable for cryptographic applications.

2.1.3. Spectral Entropy

In chaotic systems, spectral entropy (SE) is a kind of metric that characterizes the dynamics of the system from the aspect of frequency domain. SE is obtained through Fourier transforming the time series of the system to obtain the spectral distribution, and then normalizing the spectral distribution and calculating the entropy value. This value shows whether the energy distribution of the system at different frequencies is uniform or chaotic.

SE is often used to measure the degree of nonlinearity and complexity of the system. If the SE is high, the spectral distribution of the system is more uniform, the energy distribution at different frequencies is more chaotic, and the system may be in a chaotic state. Conversely, if the SE is low, it indicates that the spectral distribution of the system is more concentrated, the energy is more concentrated at certain frequencies, and the system may be in a periodic or regular state. The SE of the C-M-R model is higher, which is shown in Figure 3. At this time, the system is in a chaotic state.

2.1.4. Initial Value Sensitivity

Initial value sensitivity practically satisfies the needs of cryptography. In chaotic systems, initial value sensitivity means that the system is very sensitive to small changes in the initial conditions, i.e., a small difference in the initial value can lead to a large difference in the evolutionary trajectory of the system.

Initial value sensitivity ensures that a small change in the key or initial value will result in a completely different chaotic sequence, thus realizing the high-strength encryption of the image. Only a decryptor with the same key can correctly restore the plaintext image, while for an attacker with an unknown key, it becomes very difficult to break the encryption process. The chaotic sequences of the initial value and the initial value after scrambling interference are reflected in Figure 4, and it is seen that the chaotic sequences are different.

2.1.5. NIST Test

To better verify the suitability of chaotic systems for image encryption, chaotic sequences are verified for randomness testing. The NIST test is a commonly used test for sequence randomness and contains 15 tests. The chaotic system generates chaotic sequences of a length of 20 million and obtains 160 million binary data as data samples for the randomness test. The test results are shown in

Table 1. NIST test.

Test	p-Value	Pass Pate	Pass/Fail
The Frequency Test	0.2192	100%	Pass
Frequency Test within a Block	0.3325	98%	Pass
The Run Test	0.2627	96%	Pass
Tests for the Longest-Run-of-Ones in a Block	0.5240	98%	Pass
The Binary Matrix Rank Test	0.3165	99%	Pass
The Discrete Fourier Transform Test	0.2163	99%	Pass
The Non-Overlapping Template Matching Test	0.3197	98%	Pass
The Overlapping Template Matching Test	0.4249	99%	Pass
Universal Statistical Test	0.5414	100%	Pass
The Liner Complexity Test	0.2396	100%	Pass
The Serial Test	0.9740	100%	Pass
The Approximate Entropy Test	0.1017	100%	Pass
The Cumulative Sums Test	0.4248	100%	Pass
The Random Excursions Test	0.5881	98%	Pass
The Random Excursions Variant Test	0.6957	98%	Pass

, the p-value and pass rate. A sequence passes the randomness test when the p-value of 15 tests is greater than or equal to 0.01 and the pass rate is greater than 96%. The results show that the chaotic series passes all the tests, thus proving that the chaotic system has randomness and is suitable for image encryption algorithms.

2.2. DWT Compression

DWT is a commonly used image compression technique due to its excellent time domain analysis capability. Firstly, using the DWT technique, the image is decomposed into four sub-bands. the low-frequency sub-band is LL, and the high-frequency sub-bands are LH, HL and HH. It is worth noting that the low-frequency LL retains an approximation of the original image. Then, encryption and decryption are performed with LL. The final image is reconstructed using the LL sub-band. This DWT compression and reduction process is shown in Figure 5.

2.3. Particle Swarm Foraging Algorithm (PSFA)

The traditional particle swarm optimization algorithm finds the minimum foraging path through individuals or populations cooperating with each other. This property is utilized to disrupt the image pixels. As shown in Figure 6, four positions are randomly generated each time using the chaotic system, and after calculation, the largest direction is selected as the current moving direction. Specifically, the pixel values are traversed after selecting the farthest direction to move in.

2.4. DNA Encoding

The earliest signs of DNA are found in living organisms and consist of two complementary strands of nucleotide bases. The four nucleotide bases are A, G, C and T. They are complementary to each other, and 0 and 1 are also complementary. Therefore, the pixel values are converted into eight-bit binary values and split into combinations of 00, 01, 10, and 11 to associate with A, G, C, and T. This process of nucleotide bases from binary values is shown in Figure 7.

In the DNA coding stage, three different rules are used for the arithmetic, namely DNA addition, DNA XOR and DNA-complementary XOR (CXOR). The results are displayed in Figure 8. The color image consists of three channels, the red channel using DNA addition, the green channel adopting XOR and the blue channel utilizing CXOR.

3. Encryption Scheme and Decryption Scheme

3.1. The Complete Encryption Process

This encryption process is shown in Figure 9. First, the color image’s three channels are converted via DWT, and LL sub-bands are retained. The LL sub-bands are used in conjunction with the Hash function for assigning initial values to the chaotic system. Then, the LL sub-band is confused and diffused to generate the cipher image.

3.2. Encryption Process

3.2.1. DWT Compression

Step 1: Input a color image, T, and record its size as M × N × 3. Split the color image into T_R, T_G and T_B channels.

Step 2: Each channel undergoes DWT decomposition into four sub-bands: LL, HL, LH and HH.

$$\{\begin{array}{l}[L{L}_R,L{H}_R,H{L}_R,H{H}_R]=DWT(T_R)\\ [L{L}_G,L{H}_G,H{L}_G,H{H}_G]=DWT(T_G).\\ [L{L}_B,L{H}_B,H{L}_B,H{H}_B]=DWT(T_B)\end{array}$$

(2)

Step 3: The low-frequency sub-band information, LL, is retained and re-fused into a new matrix, T₁.

$$\{\begin{array}{l}{T}_1(:,:,1)=L{L}_R\\ T_1(:,:,2)=L{L}_G,\\ T_1(:,:,3)=L{L}_B\end{array}$$

(3)

where the T₁ size is (M/4) × (N/4) × 3.

3.2.2. Confusion Process

Step 1: The low-frequency sub-band module can be obtained via the DWT technique, and the data type of this module is not suitable for encryption. Therefore, the data need to be further processed.

Step 2: The range of values for T₁ is [min(min(T₁)), max(max(T₁))], with Equation (4) creating the range of the new T₂ value in [0, 255]:

$$T_2=\mathrm{floor}\left(\frac{T_1-\min (\min T_1)}{\max (\max T_1)-\min (\min T_1)}\right)\times 255.$$

(4)

Step 3: A rounding operation is performed on T₂ and the fractional part is discarded to obtain T₃.

$$T_3=\mathrm{floor}(T_2).$$

(5)

Step 4: T₃ is input into the Hash-256 function to obtain a set of 256-bit Hash sequences. The Hash sequences are computed to be used as initial values for the hyperchaotic system.

$$\{\begin{array}{l}{f}_1=\mathrm{mod}({10}^{-2}\times \mathrm{mean}((K(1:8)\oplus K(9:16))),L)\\ f_2=\mathrm{mod}({10}^{-2}\times \mathrm{mean}((K(17:24)\oplus K(25:32))),L)\\ f_3=\mathrm{mod}({10}^{-2}\times \mathrm{mean}((K(25:32)\oplus K(33:40))),L),\\ f_4=\mathrm{mod}({10}^{-2}\times \mathrm{mean}((K(33:40)\oplus K(41:48))),L)\\ f_5=\mathrm{mod}({10}^{-2}\times \mathrm{mean}((K(49:56)\oplus K(57:64))),L)\end{array}$$

(6)

where K represents the Hash sequences and L is the Hash length.

Step 5: The chaotic system generates x₁, y₁, z, x₂ and y₂, which are computed to obtain the desired chaotic sequence.

$$\{\begin{array}{l}{X}_1=\mathrm{floor}(\mathrm{mod}(x_1\times {10}^{12},M))+1\\ Y_1=\mathrm{floor}(\mathrm{mod}(y_1\times {10}^{12},N))+1\\ Z=\mathrm{floor}(\mathrm{mod}(z\times {10}^{12},3))+1\hspace{1em}\hspace{1em},\\ X_2=\mathrm{floor}(\mathrm{mod}(x_2\times {10}^{12},M))+1\\ Y_2=\mathrm{floor}(\mathrm{mod}(y_2\times {10}^{12},N))+1\end{array}$$

(7)

where x₁ and x₂ are random lengths, and y₁ and y₂ are random widths. z is the parameter required to span the channel.

Step 6 (the PSFA stage): For the traversal of the pixel values, each point of the pixel value has a total of four random points to be exchanged. Thus, the furthest-distance pixel point is selected.

$$\{\begin{array}{l}\mathrm{fitness}(1)={(k_x-X_1(k_x,k_y,k_z))}^2+{(k_y-Y_1(k_x,k_y,k_z))}^2\\ \mathrm{fitness}(2)={(k_x-X_1(k_x,k_y,k_z))}^2+{(k_y-Y_2(k_x,k_y,k_z))}^2\\ \mathrm{fitness}(3)={(k_x-X_2(k_x,k_y,k_z))}^2+{(k_y-Y_1(k_x,k_y,k_z))}^2,\\ \mathrm{fitness}(4)={(k_x-X_2(k_x,k_y,k_z))}^2+{(k_y-Y_2(k_x,k_y,k_z))}^2\\ [\mathrm{index}]=\max (\mathrm{fitness})\end{array}$$

(8)

Step 7: It is worth noting that the results of each traversal may swap pixel values in different channels. This choice is selected by the chaotic sequence. Equation (9) represents four ways to make this choice. Applying the rules of Equations (7) and (9), the results are shown in Figure 10.

$$\{\begin{array}{l}t=T_1(k_x,k_y,k_z)\\ T_1(k_x,k_y,k_z)=T_1(X(k_x,k_y,k_z),Y(k_x,k_y,k_z),Z(k_x,k_y,k_z)).\\ T_1(k_x,k_y,k_z)=t\end{array}$$

(9)

3.2.3. Diffusion Process

Step 1: The confused image, T₃, is converted to 8-bit binary values via data type conversion.

$$\{\begin{array}{l}TR=\mathrm{dec}2\mathrm{bin}(T_3(:,:,1)\\ TG=\mathrm{dec}2\mathrm{bin}(T_3(:,:,2).\\ TB=\mathrm{dec}2\mathrm{bin}(T_3(:,:,3)\end{array}$$

(10)

Step 2: Data type conversion is performed on chaotic sequences as well.

$$\{\begin{array}{l}{H}_1=\mathrm{dec}2\mathrm{bin}(X_1(:,:,1)\\ H_2=\mathrm{dec}2\mathrm{bin}(X_1(:,:,2).\\ H_3=\mathrm{dec}2\mathrm{bin}(X_1(:,:,3)\end{array}$$

(11)

Step 3: Three rules of DNA operation are used in Figure 2 that are used for the R channel; the XOR rule is used for the G channel and the CXOR rule is used for the B channel. The first rule in Figure 1 is chosen as the DNA encoding rule, and the encryption process is indicated in Figure 11.

$$\{\begin{array}{l}TR=TR+H_1\\ TG=TG\oplus H_2.\\ TB=\overline{TB\oplus H_3}\end{array}$$

(12)

3.3. Decryption Process

The decryption process performs inverse diffusion and inverse confusion operations. Firstly, the chaotic key is input into the chaotic system to obtain the chaotic sequence needed for decryption. Second, inverse diffusion is performed, including the DNA decoding process. Then, inverse disarray is performed. Finally, the image is recovered with the use of the IDWT technique. A detailed decryption flowchart is presented in Figure 12.

3.4. Simulation Results

This article makes use of ‘Peppers’ (256 × 256 × 3), ‘Fruits’ (512 × 512 × 3) and ‘2.2.05′ (1024 × 1024 × 3) color images. The simulation results are indicated in Figure 13. The simulation result shows that the restored image is identical to the plaintext image except for the image brightness.

4. Security Analysis

4.1. Quality Reconstruction Analysis

The DWT technique is used to decompose the color image into bands of different scales. The LL band retains the general outline and main structure of the original image. On the decryption side, IDWT reduction of the image is performed. However, lossy compression brings about a loss of quality in the image. PSNR is a common measure of image quality and is usually measured in decibels (dB). The higher the PSNR, the better the image is recovered. The PSNR test is shown in

Table 2. PSNR test.

Peppers	Size	PSNR	Fruits	Size	PSNR	2.2.05	Size	PSNR
R	256 × 256	29.5744	R	512 × 512	33.0693	R	1024 × 1024	29.2794
G	256 × 256	26.6075	G	512 × 512	32.3118	G	1024 × 1024	30.0306
B	256 × 256	29.3594	B	512 × 512	32.5318	B	1024 × 1024	31.6912

. The PSNR is calculated as in Equation (13).

$$\mathrm{PSNR}=10\lg \frac{M\times N\times {255}^2}{{\sum }_{i=1}^M{\sum }_{j=1}^N|T(i,j)-Y(i,j){|}^2}.$$

(13)

There are around 30 PSNR test results, which are compared with the results of other compression methods using ‘Peppers’ images with a size of 256 × 256, as shown in

Table 3. Comparison of DWT compression with other references.

Scheme	[53]	[54]	[55]	[56]	Proposed
PSNR (dB)	26.2896	27.6917	27.3366	27.4886	28.5137

.

4.2. Histogram Analysis

Histograms are the basis of statistical analysis and reflect the number of pixel values per pixel in an image. Some pixels in the original image are particularly focused, while others have a very small number of pixel values, exposing the pixel value characteristics of the original image. Normally, the histogram of an encrypted image should show a uniform distribution with no obvious peaks or features to ensure that the encrypted image does not leak information from the plaintext image.

The histogram pixel values of the plaintext image with uneven features, and the histogram pixels of the cipher image with uniform pixels, shown in Figure 14, indicate that the attacker cannot analyze the cipher image to obtain information about the features of the pixels of the normal image.

4.3. Correlation Analysis

In plaintext images, the correlation between adjacent pixel values is high, and an attacker can obtain the image by analyzing the correlation. In contrast, the original image id encrypted so that the correlation between its pixels is destroyed, and there is essentially no correlation between the pixel values in the encrypted image.

The neighboring pixels in three different directions of the original grayscale image, ‘2.2.05′, are decomposed and the neighboring pixels, as shown in Figure 15. The correlation coefficient of the neighboring pixels of the image is calculated using the following equation.

$$r_{x,y}=\frac{E[(x-E(x))(y-E(y))]}{{\sigma }_x{\sigma }_y}$$

(14)

where E(x) is the expected value of x and σ_x is the standard deviation of x.

Table 4. Correlation coefficient of the ‘Peppers’ image and its cipher’s neighboring pixels.

Peppers	Horizontal	Vertical	Diagonal	Pass/Fail
R	0.8780	0.8559	0.8406	Fail
G	0.8542	0.8863	0.8109	Fail
B	0.8247	0.8572	0.8082	Fail
Cipher	Horizontal	Vertical	Diagonal	Pass/Fail
R	0.0029	0.0023	−0.0061	Pass
G	0.0092	−0.0107	0.0015	Pass
B	−0.0033	0.0025	−0.0008	Pass

lists the different correlation coefficients for the three channels of the color image ‘Peppers’. From the table, it can be concluded that the encrypted neighbor correlation coefficients are close to 0 and can effectively resist statistical attacks.

4.4. Information Entropy

Information entropy (IE) is a measure of image complexity and information content. IE describes the probability of each pixel appearing. The theoretical value of IE is usually considered to be 8 [57], and the IE of this test is very close to the theoretical value. The IE value of the encrypted image should be high enough to indicate that the information in the image is uniformly and randomly spread so that an attacker cannot derive useful information from it. The results of this information entropy is shown in

Table 5. Original image and cipher image information entropy.

Images		Plain Image				Cipher Image
	Size	R	G	B	Size	R	G	B
Peppers	256 × 256	7.3318	7.5242	7.0792	128 × 128	7.9901	7.9887	7.9889
Fruits	512 × 512	7.0555	7.3527	7.7134	256 × 256	7.9968	7.9968	7.9969
2.2.05	1024 × 1024	7.4445	7.2346	6.5895	512 × 512	7.9992	7.9992	7.9993

.

4.5. Key Space

The size of the key space determines the resistance of the encryption scheme to exhaustive attacks. When the key space is larger than 2¹⁰⁰ [58], it can resist exhaustive attacks, and the larger the key space, the smaller the probability that the encryption scheme will be cracked. By slightly perturbing each parameter in the decryption process, the key space of each parameter can be obtained, and the total key space can be obtained after calculation.

Table 6. Key space.

Parameters	Key Space
c, γ, σ, f2, f3	1015
ε, kr, a, b, μ, α, β, f1, f2, f5	1016
Total key space	10235 ≈ 2780

shows the key space, and

Table 7. Key spaces of different schemes.

Scheme	Ref. [59]	Ref. [60]	Ref. [61]	Proposed
Key space	2192	2256	2358	2780

shows the references for it, which show that our key space is much larger and resistant to brute-force cracking.

4.6. Key Sensitivity

To test the key sensitivity performance, a color image of ‘2.2.05′ is used. Encryption is performed using deterministic parameters, and its encryption result is T₁. Then, scrambling is performed on each parameter separately, and its encryption result is T₂. The difference between the T₁ image and T₂ image is evaluated by the rate of change in pixel values (NPCR) of the two images (normally encrypted and scrambled, encrypted images). The scheme has high key sensitivity, and the NPCR results are shown in

Table 8. NPCR with different parameters.

Key Space	Δ Value	NPCR (%)
Ε	Δ = 10−16	99.6052
kr	Δ = 10−16	99.6033
a	Δ = 10−16	99.6113
b	Δ = 10−16	99.6193
c	Δ = 10−15	99.6048
r	Δ = 10−15	99.6155
μ	Δ = 10−16	99.6124
σ	Δ = 10−15	99.6231
α	Δ = 10−16	99.6243
β	Δ = 10−16	99.6185
f1	Δ = 10−16	99.6075
f2	Δ = 10−15	99.6123
f3	Δ = 10−15	99.6105
f4	Δ = 10−16	99.6090
f5	Δ = 10−16	99.5975
Average value	Δ = 10−15	99.6116

.

$$\mathrm{NPCR}(T_1,T_2)=\frac{1}{MN}{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N\left|\mathrm{Sign}\left(T_1\left(i,j\right)-T_2\left(i,j\right)\right)\right|\times 100\%}}.$$

(15)

where Sign(•) is a symbolic function.

In the test, the specifications of the color images ‘Peppers’ and ‘2.2.05′ are encrypted with the normal key and revised with one key, respectively; the results are shown in Figure 16.

4.7. Differential Attack

Differential attacks are a common method for attackers to use to break algorithms. The attacker randomly changes one pixel point of the original image to obtain the encrypted image and analyzes the difference between the two cipher images to break the scheme.

In the differential attack test, the original image is encrypted twice; the first time it is encrypted involves normal encryption, T₁, and the second time, the attacker randomly changes one pixel point of the original image to obtain T₂. Since the scheme’s plaintext information is associated with the hash function, randomly changing one pixel of the original image will, in turn, result in a different hash sequence. This leads to a change in the initial value of the C-M-R system, and its chaotic sequence, in turn, changes. As a result, the encrypted structure and content change, and in turn, the encrypted image changes.

The difference between T₁ and T₂ is evaluated using the NPCR and the intensity of change (UACI) of the two images. The results of the tests are indicated in

Table 9. Test results of different images.

Images	NPCR (%)	UACI (%)
‘Peppers’	99.5850	33.3926
‘Fruits’	99.6017	33.4460
‘2.2.05’	99.6235	33.4601
Proposed	99.6034	33.4329

. As can be seen from

Table 10. Differential attacks of different schemes.

Schemes	NPCR (%)	UACI (%)
Ref. [62]	99.57	33.58
Ref. [63]	99.6060	33.5126
Ref. [64]	99.6231	33.4162
Proposed	99.6034	33.4329

, the percentage change of pixels of this scheme is better than that of other reference schemes.

$$\mathrm{UACI}\left(T_1,T_2\right)=\frac{1}{MN}{\displaystyle \sum _{i=0}^M{\displaystyle \sum _{j=0}^N\frac{\left|T_1\left(i,j\right)-T_2\left(i,j\right)\right|}{255-0}\times 100\%}}.$$

(16)

4.8. Plain Attack

It is well known that there are four classic types of attacks: cipher attacks, known plaintext attacks, chosen plaintext attacks and chosen cipher attacks. Among them, chosen plaintext attacks are the most powerful attacks. If a cryptosystem is able to withstand these, it can resist other types of attacks.

Due to the specificity of the DWT technique, here, half of the all-black and half of the all-white images are selected, and this result is shown in Figure 17; Figure 17c after this encryption scheme is generated as a noise-like image, which shows that this encryption scheme is able to resist plaintext attacks.

4.9. Robustness

4.9.1. Noise Attack Analysis

As encrypted images are transmitted over the network, they are mostly interfered with by various noises in the channel, such as salt-and-pepper noise (SPN) and Gaussian noise (GN), which makes the recovery of ordinary images more difficult. In this section, the effectiveness of the algorithm against noise attacks is described. Specifically, the color image ‘Fruits’ is used as the decrypted images in the case of ordinary images with test noise. Therefore, the algorithm is robust against noise to a certain extent in this article. The results of this noise test is shown in Figure 18.

4.9.2. Shearing Attack Analysis

A shear attack is an attack that cuts against the pixels of an image. An attacker may cut and crop specific portions to gain access to sensitive information or to break encryption algorithms. The encrypted image for this test is the color image ‘Fruits’, and the cipher image is sheared with 6.25%, 12.5%, and 25% shearing attack to regain the cipher image. The recovered image can be seen in the figure. Thus, this algorithm has some resistance against shearing attacks. The result of this shearing attack is shown in Figure 19.

4.10. Comparing Our Scheme with Other Encryption Schemes

In summary, various performance metrics discussed above are considered to compare the proposed encryption scheme with other state-of-the-art chaotic and nonchaotic encryption schemes in

Table 11. Comparing our scheme with other encryption schemes.

Process	Proposed	[65]	[66]	[67]	[68]
Key Space	2780	2149	-	1075	28 × 64 × 64 × 3 × 10
NPCR (%)	99.6034	99.5985	99.6148	-	99.5994
UACI (%)	33.4329	33.4388	33.4268	-	33.4186
Entropy	7.9993	-	7.9993	7.9994	-

. Refs. [65,66,67] present chaotic encryption schemes. Ref. [68] uses adaptive Fourier decomposition (AFD) for image encryption. When comparing the proposed DWT-compressed encryption scheme based on chaos theory with the above schemes, it can be concluded that our scheme is superior.

5. Conclusions

An encryption scheme with discrete memristive coupled neurons and DWT compression is introduced in this article. The behavior dynamics of chaotic systems are analyzed by means of phase diagrams, LEs, bifurcation diagrams and SE. The color image is compressed via DWT, and the parameters related to the color image information are obtained using the Hash function, i.e., a set of plaintexts corresponds to a set of keys. The pseudo-random sequence obtained using the chaotic system is used for encryption. The pixel positions of the color image are exchanged using chaotic sequences as a tool guided by the movement rules of the PSFA. DNA encoding rules are utilized to further secure the image. Simulation results show that the DWT technique and encryption scheme can compress color images of different sizes into a quarter of the original image capacity and encrypt them into unrecognizable images. The security analysis shows that the C-M-R discrete map introduces a large key space to defend the scheme against exhaustive attacks. Histogram, correlation and information entropy statistics show that the PSFA and DNA encoding can withstand various statistical analyses. Differential attack analysis shows the superiority of our algorithms in terms of security performance. The encrypted image retains its ability for quality recovery even when the encrypted image is subjected to external disturbances such as malicious clipping and noise contamination.

In future work, as DWT compression can only compress the image into a quarter of the original image, future improvements in the DWT compression technique are required to fulfill the arbitrary compression ratio of the image. Moreover, the next step is to design a new encryption scheme to satisfy image encryption, as the DNA encoding process is time-consuming.